14 2. A Summary of Basic Determinant Theory
=
1
A
n
∑
i=1
biAij. (2.3.14)
The solution of the triangular set of equations
i
∑
j=1
aijxj=bi,i=1, 2 , 3 ,...
(the upper limit in the sum isi, notnas in the previous set) is given by
the formula
xi=
(−1)
i+1
a 11 a 22 ···aii
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣
b 1 a 11
b 2 a 21 a 22
b 3 a 31 a 32 a 33
··· ··· ··· ··· ··· ···
bi− 1 ai− 1 , 1 ai− 1 , 2 ai− 1 , 3 ··· ai− 1 ,i− 1
bi ai 1 ai 2 ai 3 ··· ai,i− 1
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣
i
.
(2.3.15)
The determinant is a Hessenbergian (Section 4.6).
Cramer’s formula is of great theoretical interest and importance in solv-
ing sets of equations with algebraic coefficients but is unsuitable for reasons
of economy for the solution of large sets of equations with numerical coeffi-
cients. It demands far more computation than the unavoidable minimum.
Some matrix methods are far more efficient. Analytical applications of
Cramer’s formula appear in Section 5.1.2 on the generalized geometric se-
ries, Section 5.5.1 on a continued fraction, and Section 5.7.2 on the Hirota
operator.
Exercise.If
f
(n)
i
=
n
∑
j=1
aijxj+ain, 1 ≤i≤n,
and
f
(n)
i
=0, 1 ≤i≤n, i=r,
prove that
f
(n)
r
=
Anxr
A
(n)
rn
, 1 ≤r<n,
f
(n)
n =
An(xn+1)
An− 1
,
where
An=|aij|n,
provided
A
(n)
rn =0,^1 ≤i≤n.